@tao @christianp I have a problem for you in Number Theory

The Bertrand postulate which states that there is a prime between (n, 2n) for all n > 2. I wanted to improve this result.

So I defined \( \delta(n) = \sum_i p_i \) for \( n = \prod_i p_i \) the prime descomposition of n.

I want to prove that for all n, there is a prime between \( n, n + \delta(n) \) which is false: 5120 is the first of such number which has no primes in this interval.

Let's define a "refinate number" a number which has prime in the interval \( n, n + \delta(n) \) and "irrefinate" else.

I want to prove that irrefinate number are very "rare".

As a consequence of Prime Number Theorem, we have that for fixed m, mp for p prime large enough.

And if irrefinate numbers are infinite, then \( \delta(n)/n \) tends to 0.

But I don't know nothing.

It could be interesting knowing the counting function of irrefinate is.

Link to unsolved problems: "for all n, there is a prime number between \( n^2 , (n+1)^2 \) .

In general, \( \delta(n^2) < 2n +1 \) but I don't know how frequently.

It connects #addivite function to #prime #number #gaps.

It was just a divertimento in my graduate days....

Thanks in advance and regards